Final answer:
The Rational Root Theorem dictates that possible rational roots of the function f(x)=3x^2+2x-1 are ±1 and ±1/3. After testing each possible root with synthetic division and finding no remainder of zero, we conclude that there are no rational roots and alternative methods should be used to solve the equation.
Step-by-step explanation:
To find the possible rational roots of the function f(x)=3x^2+2x−1, we can apply the Rational Root Theorem. The theorem states that any rational root, expressed in its simplest form as p/q, is such that p is a factor of the constant term (in this case -1) and q is a factor of the leading coefficient (which is 3 for the given function).
The factors of -1 are ±1, and the factors of 3 are ±1 and ±3. Therefore, the possible rational roots are ±1, ±1/3. To test these possible roots using synthetic division, we set up a synthetic division table for each potential root and perform the operation. If the remainder is 0, the tested value is a root of the function.
However, after performing synthetic division with each possible root, we find that there is no rational root that gives a remainder of zero. Therefore, the quadratic equation f(x)=3x^2+2x−1 does not have a rational root, and we might need to resort to other methods such as the quadratic formula or numerical methods to solve it.